Properties

Label 3724.113
Modulus $3724$
Conductor $931$
Order $14$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(14))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,7]))
 
pari: [g,chi] = znchar(Mod(113,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{931}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.bv

\(\chi_{3724}(113,\cdot)\) \(\chi_{3724}(645,\cdot)\) \(\chi_{3724}(1709,\cdot)\) \(\chi_{3724}(2241,\cdot)\) \(\chi_{3724}(2773,\cdot)\) \(\chi_{3724}(3305,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.171249048453908271645616513339.1

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{5}{7}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{9}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3724 }(113,a) \;\) at \(\;a = \) e.g. 2